On regular graphs equienergetic with their complements
| Autor: | Podestá, Ricardo A., Videla, Denis E. |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We give necessary and sufficient conditions on the parameters of a regular graph $\Gamma$ (with or without loops) such that $E(\Gamma)=E(\overline \Gamma)$. We study complementary equienergetic cubic graphs obtaining classifications up to isomorphisms for connected cubic graphs with single loops (5 non-isospectral pairs) and connected integral cubic graphs without loops ($\Gamma = K_3 \square K_2$ or $Q_3$). Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs $Cr(n)$ or $C_4$. Next, for the family of strongly regular graphs $\Gamma$ we characterize all possible parameters $srg(n,k,e,d)$ such that $E(\Gamma) = E(\overline \Gamma)$. Furthermore, using this, we prove that a strongly regular graph is equienergetic to its complement if and only if it is either a conference graph or else it is a pseudo Latin square graph (i.e. has $OA$ parameters). We also characterize all complementary equienergetic pairs of graphs of type $\mathcal{C}(2)$, $\mathcal{C}(3)$ and $\mathcal{C}(5)$ in Cameron's hierarchy (the cases $\mathcal{C}(1)$ and $\mathcal{C}(4)$ are still open). Finally, we consider unitary Cayley graphs over rings $G_R=X(R,R^*)$. We show that if $R$ is a finite Artinian ring with an even number of local factors, then $G_R$ is complementary equienergetic if and only if $R=\mathbb{F}_q \times \mathbb{F}_{q'}$ is the product of 2 finite fields. Comment: Some additions, the paper grow from25 to 32 pages (5 tables). We add arbitrary number of loops in Proposition 2.3 and some examples with graphs with loops. Cubic graphs are now in a section (3). In section 8 we add "Unitary Cayley graphs with loops". Final remarks added |
| Databáze: | arXiv |
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